1135 GMT January 22, 2018
Joshua, 28, holds the IBF and WBA belts, while New Zealander Parker is the WBO champion, BBC reported.
Cardiff's Principality stadium – where Joshua successfully defended his titles against Carlos Takam in October – will host the bout.
"I'm delighted to get this fight made – it's been a long time coming," said Joshua's promoter Eddie Hearn.
"It's the first time in history that two reigning heavyweight world champions have met in Britain."
Joshua added, "These fights aren't easy because there is a lot on the line – so respect to team Parker for taking the challenge.
"I'm looking forward to it. Training camp is under way and before you know, March 31 will be upon us."
Parker's representative David Higgins arrived in the UK last week to finalize a deal. Talks had previously stalled over how the purse would be split but a compromise was reached last month which will see Parker, 25, earn 30-35 percent.
Parker is undefeated in his 24 professional fights, with 18 of those wins coming by a knockout.
"Anthony Joshua is in for a huge shock," Parker said.
"A couple of months ago I heard him say: 'Why should I be worried about this little kid from New Zealand?'
"Well, now he's about to find out. And the world is about to find out whether AJ can really take a punch."
Joshua became IBF heavyweight champion by defeating Charles Martin at London's O2 Arena in April 2016.
The following April, he added the WBA belt by beating Wladimir Klitschko in front of 90,000 fans at Wembley Stadium.
Whoever wins in March is certain to then face undefeated American Deontay Wilder, who holds the fourth belt in the heavyweight division. No boxer has held all four heavyweight belts simultaneously.
Parker's camp have been pushing for the fight ever since Joshua defeated Takam last year.
They staged a news conference in November featuring clips of Joshua being knocked down in his amateur days and offered £20,000 to anyone with footage of him being floored.
That prompted Joshua to hit back, saying he was being "demonized" by the Parker camp. But Higgins later told BBC Sport Joshua was "rattled".
Parker, who will fly to Las Vegas for nine weeks of training, told Radio 5 live that his news conference "played a big part in making the fight happen".
"He got angry and upset,” said Parker.
"Everything along the way helped, now we can put that behind us and get training."
Parker said he and Joshua have found "common ground" over the purse adding that he "respects" the Londoner.
"There is no animosity," added the Kiwi.
"I am a relaxed, calm person. I have respect for Joshua, you do not have a 20-0 record if you are not that good.
"But I think I can beat anyone in the world, I would not be here if I did not believe in myself."
On unifying the division, he added, "The world needs one champion."